Probability of finding a particle and complex wave functions

[Maroon Ray]

1-Why is the probability of finding a particle at
specific position at a certain time
proportional to |Ψ|2?
2- Waves are represented by sinusoidal or
complex functions. Why did we choose a
complex function to represent matter waves
instead of sinusoidal?

 

[masudr]

1-Why is the probability of finding a particle at
specific position at a certain time
proportional to |Ψ|2?

You are questioning a postulate; a postulate cannot be explained in the same framework. It ought to be taken as given.

2- Waves are represented by sinusoidal or
complex functions. Why did we choose a
complex function to represent matter waves
instead of sinusoidal?

The solution of the Schrodinger equation (which defines what can be allowed as wavefunctions) forces us to use complex functions. You can question the validity of the Schrodinger equation, but it is too a postulate, and so the same argument above applies.

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So I cannot answer why the above hold within QM, but it can be made plausible. The first point is that all these calculations that agree remarkably well depend upon both of the above postulates. And then there are a whole host of other plausibility arguments based on the classical limit.

 

[quantumdude]

2- Waves are represented by sinusoidal or
complex functions. Why did we choose a
complex function to represent matter waves
instead of sinusoidal?

Are you asking why the solutions of the Schrodinger equation are complex, or why we write them in terms of complex exponentials? If it's the first one then see masudr's reply: the solutions are what they are, and we can't do anything about them. If your question is the second one then the answer is that the basis [itex]{\sin(kx),\cos(kx)}[/itex] spans the same vector space as the basis [itex]{exp(ikx),exp(-ikx)}[/itex]. We can arbitrarily switch between them.